Newton's method of estimation - using derivatives

Newton devised the following method for approximating a real root of the equation f(x) = 0. i.e. a real number for which f(r) = 0. We begin by guessing an approximation, say x1, to the real root r.

(i) Find the equation of the line tangent to the graph of y = f(x) at the point (x1,f(x1)). Assume f is differentiable and f’(x1) ≠ 0.

(ii) Newton felt that the x-intercept of the tangent line in part(i) would be a better approximation to r than x1. Label the point of intersection of the tangent line in (i) with the x-axis (x2,0). Find x2 in terms of x1, f(x1), and f’(x1).

(iii) The above procedure may now be repeated using the tangent line at (x2,f(x2)). If f’(x2) ≠ 0 this leads to a third approximation, x3, where (x3,0) is the point of intersection of the x-axis with the tangent line at (x2,f(x2)). Find x3 in terms of x2, f(x2), and f’(x2).

If done correctly, you have just derived Newton’s method, which can be written as:
To approximate the real root of r of f(x)=0, begin by guessing an initial approximation to r, say x1, For n = 1, 2, 3, … let xn+1 = xn – f(xn)/f’(xn). Under “good” conditions the sequence xn will converge to r.

(iv) Approximate k^.5 for k > 0 by applying Newton’s method to f(x) = x^2 – k with the initial guess x1, You have just derived the method Heron used around 100 A.D. to approximate k^.5.